Alliances in Directed Graphs

Alliances in undirected graphs were introduced by Hedetniemi, Hedetniemi, and Kristiansen, and generalized to \( k \)-alliances by Shafique and Dutton. We translate these definitions of alliances to directed graphs. We establish basic properties of alliances and examine bounds on the size of minimal alliances in directed graphs. In general, the bounds established for alliances in undirected graphs do not hold when alliances are considered over the larger class of directed graphs and we construct examples which break these bounds.